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Theorem notzfausOLD 5256
Description: Obsolete proof of notzfaus 5255 as of 18-Nov-2023. (Contributed by NM, 8-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
notzfaus.1 𝐴 = {∅}
notzfaus.2 (𝜑 ↔ ¬ 𝑥𝑦)
Assertion
Ref Expression
notzfausOLD ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem notzfausOLD
StepHypRef Expression
1 notzfaus.1 . . . . . 6 𝐴 = {∅}
2 0ex 5204 . . . . . . 7 ∅ ∈ V
32snnz 4705 . . . . . 6 {∅} ≠ ∅
41, 3eqnetri 3086 . . . . 5 𝐴 ≠ ∅
5 n0 4310 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
64, 5mpbi 232 . . . 4 𝑥 𝑥𝐴
7 biimt 363 . . . . . 6 (𝑥𝐴 → (𝑥𝑦 ↔ (𝑥𝐴𝑥𝑦)))
8 iman 404 . . . . . . 7 ((𝑥𝐴𝑥𝑦) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝑥𝑦))
9 notzfaus.2 . . . . . . . 8 (𝜑 ↔ ¬ 𝑥𝑦)
109anbi2i 624 . . . . . . 7 ((𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ¬ 𝑥𝑦))
118, 10xchbinxr 337 . . . . . 6 ((𝑥𝐴𝑥𝑦) ↔ ¬ (𝑥𝐴𝜑))
127, 11syl6bb 289 . . . . 5 (𝑥𝐴 → (𝑥𝑦 ↔ ¬ (𝑥𝐴𝜑)))
13 xor3 386 . . . . 5 (¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ (𝑥𝑦 ↔ ¬ (𝑥𝐴𝜑)))
1412, 13sylibr 236 . . . 4 (𝑥𝐴 → ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)))
156, 14eximii 1833 . . 3 𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑))
16 exnal 1823 . . 3 (∃𝑥 ¬ (𝑥𝑦 ↔ (𝑥𝐴𝜑)) ↔ ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑)))
1715, 16mpbi 232 . 2 ¬ ∀𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
1817nex 1797 1 ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1531   = wceq 1533  wex 1776  wcel 2110  wne 3016  c0 4291  {csn 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-nul 5203
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3497  df-dif 3939  df-nul 4292  df-sn 4562
This theorem is referenced by: (None)
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